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        "%matplotlib inline"
      ]
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      "source": [
        "\n# Kernel Density Estimation\n\n\nThis example shows how kernel density estimation (KDE), a powerful\nnon-parametric density estimation technique, can be used to learn\na generative model for a dataset.  With this generative model in place,\nnew samples can be drawn.  These new samples reflect the underlying model\nof the data.\n\n"
      ]
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    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
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      "source": [
        "import numpy as np\nimport matplotlib.pyplot as plt\n\nfrom sklearn.datasets import load_digits\nfrom sklearn.neighbors import KernelDensity\nfrom sklearn.decomposition import PCA\nfrom sklearn.model_selection import GridSearchCV\n\n# load the data\ndigits = load_digits()\n\n# project the 64-dimensional data to a lower dimension\npca = PCA(n_components=15, whiten=False)\ndata = pca.fit_transform(digits.data)\n\n# use grid search cross-validation to optimize the bandwidth\nparams = {'bandwidth': np.logspace(-1, 1, 20)}\ngrid = GridSearchCV(KernelDensity(), params)\ngrid.fit(data)\n\nprint(\"best bandwidth: {0}\".format(grid.best_estimator_.bandwidth))\n\n# use the best estimator to compute the kernel density estimate\nkde = grid.best_estimator_\n\n# sample 44 new points from the data\nnew_data = kde.sample(44, random_state=0)\nnew_data = pca.inverse_transform(new_data)\n\n# turn data into a 4x11 grid\nnew_data = new_data.reshape((4, 11, -1))\nreal_data = digits.data[:44].reshape((4, 11, -1))\n\n# plot real digits and resampled digits\nfig, ax = plt.subplots(9, 11, subplot_kw=dict(xticks=[], yticks=[]))\nfor j in range(11):\n    ax[4, j].set_visible(False)\n    for i in range(4):\n        im = ax[i, j].imshow(real_data[i, j].reshape((8, 8)),\n                             cmap=plt.cm.binary, interpolation='nearest')\n        im.set_clim(0, 16)\n        im = ax[i + 5, j].imshow(new_data[i, j].reshape((8, 8)),\n                                 cmap=plt.cm.binary, interpolation='nearest')\n        im.set_clim(0, 16)\n\nax[0, 5].set_title('Selection from the input data')\nax[5, 5].set_title('\"New\" digits drawn from the kernel density model')\n\nplt.show()"
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